# Semi-local ring

In mathematics, a **semi-local ring** is a ring for which *R*/J(*R*) is a semisimple ring, where J(*R*) is the Jacobson radical of *R*. (Lam & 2001, §20)(Mikhalev & 2002, C.7)

The above definition is satisfied if *R* has a finite number of maximal right ideals (and finite number of maximal left ideals). When *R* is a commutative ring, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many maximal ideals".

Some literature refers to a commutative semi-local ring in general as a
*quasi-semi-local ring*, using semi-local ring to refer to a Noetherian ring with finitely many maximal ideals.

A semi-local ring is thus more general than a local ring, which has only one maximal (right/left/two-sided) ideal.

## Examples

- Any right or left Artinian ring, any serial ring, and any semiperfect ring is semi-local.
- The quotient is a semi-local ring. In particular, if is a prime power, then is a local ring.
- A finite direct sum of fields is a semi-local ring.
- In the case of commutative rings with unity, this example is prototypical in the following sense: the Chinese remainder theorem shows that for a semi-local commutative ring
*R*with unit and maximal ideals*m*_{1}, ..., m_{n}

- .
- (The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩
_{i}m_{i}=J(*R*), and we see that*R*/J(*R*) is indeed a semisimple ring.

- The classical ring of quotients for any commutative Noetherian ring is a semilocal ring.
- The endomorphism ring of an Artinian module is a semilocal ring.
- Semi-local rings occur for example in algebraic geometry when a (commutative) ring
*R*is localized with respect to the multiplicatively closed subset*S = ∩ (R \ p*, where the_{i})*p*are finitely many prime ideals._{i}

## Textbooks

- Lam, T. Y. (2001), "7",
*A first course in noncommutative rings*, Graduate Texts in Mathematics,**131**(2 ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439 - Mikhalev, Alexander V.; Pilz, Günter F., eds. (2002),
*The concise handbook of algebra*, Dordrecht: Kluwer Academic Publishers, pp. xvi+618, ISBN 0-7923-7072-4, MR 1966155